The Department of Mathematical Sciences is hosting a workshop devoted to connections between knot theory and quantum computing. It will be the first conference in a series of planned conferences held at UT Dallas about relations between knot theory and other fields of sciences.

Knot theory is the mathematical branch of topology that studies mathematical knots, which are defined as embeddings of a circle in 3-dimensional space (imagine a knotted string with the ends joined together). Quantum computing involves the direct use of the relationship between energy and matter to perform operations on data.

ABSTRACT: In recent years there have been exciting new developments in the area of knot theory and 3-manifold topology and quantum computing. From knot invariants, quantum invariants of 3-manifolds, and topological quantum field theories, new connections between knot theory and quantum computing were established by L. Kauffman and S. Lomonaco.

Temperley-Lieb Recoupling Theory (TL-Recupling Theory) proposed by Kauffman generalizes the standard angular momentum recoupling theory and Penrose theory of spin netwarks. In a paper published in 2000, Khovanov introduces a "categorification" process to obtain invariants using homological algebra. For each link, he defines a series of homology groups which, when computing Euler characteristic, yield the Jones polynomial.

These homology groups turn out to be surprisingly strong and have been used to solve a number of well-known problems in topology. Khovanov conjectures that his idea of categorification can be used to unify all the theories mentioned above in dimensions three and four. Recent work of Ozsvath-Szabo establishes a direct connection between Khovanov theory and Seiberg-Witten-Floer theory via spectral sequence.

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